344 Chapter 9. Cooperative transitions in macromolecules[[Student version, January 17, 2003]]
In this formula,f ̄=fA/kBT.AdjustingAto fit the experimental data gave the solid black curve
shown in Figure 9.4.
9.5.1′The angle of optical rotation is not intrinsic to the molecular species under study: It depends
on the concentration of the solution and so on. To cure this defect, biophysical chemists define the
“specific optical rotation” as the rotation angleθdivided byρmd/(100kg·m−^2 ), whereρmis the
mass concentration of the solute anddis the path length through solution traversed by the light.
The data in Figure 9.6 show specific optical rotation (at the wavelength of the sodium D-line).
With this normalization, the three different curves effectively all have the same total concentration
of monomers, and so may be directly compared.
9.5.3′
- Our discussion focused upon hydrogen-bonding interactions between monomers in a polypep-
tide chain. Various other interactions are also known to contribute to the helix–coil transition,
for example dipole-dipole interactions. Their effects can be summarized in the values of the phe-
nomenological parameters of the transition, which we fit to the data. - The analysis of Section 9.5.2 did not take into account the polydispersity of real polymer samples.
Wecan make this correction in a rough way as follows.
Suppose a sample contains a numberXj of chains of lengthj. Then the fraction isfj =
Xj/(
∑
kXk), and thenumber-averaged chain lengthis defined asN#≡∑
j(jfj). Another
kind of average can also be determined experimentally, namely theweight-averaged chain length
Nw≡(1/N#)
∑
j(j^2 fj).
Zimm, Doty and Iso quoted the values (N#=40,Nw=46) and (N#=20,Nw=26) for their
twoshort-chain samples. Let us model these samples as each consisting oftwoequal subpopulations,
of lengthskandm.Then choosingk=55,m=2 4 reproduces the number- and weight-averaged
lengths of the first sample, and similarlyk=31,m=9for the second one. The lower two curves in
Figure 9.6 actually show a weighted average of the result following from Your Turn 9l(c), assuming
the two subpopulations just mentioned. Introducing the effect of polydispersity, even in this crude
way,does improve the fit to the data somewhat.
9.6.3′ Austin and coauthors obtained the fits shown in Figure 9.12 as follows. Suppose that each
conformational substate has a rebinding rate related to its energy barrierE‡byan Arrhenius rela-
tion,k(E‡,T)=Ae−E‡/kBT.The subpopulation in a given substate will relax exponentially, with
this rate, to the bound state. We assume that the prefactorAdoes not depend strongly on tem-
perature in the range studied. Letg(E‡)dEdenote the fraction of the population in substates with
barrier betweenE‡andE‡+dE.Wealso neglect any temperature dependence in the distribution
functiong(E‡).
The total fraction in the unbound state at time t will then be N(t, T)=
N 0
∫
dE‡g(E‡)e−k(E‡,T)t. The normalization factor N 0 is chosen to get N =1at time
zero. Austin and coauthors found that they could fit the rebinding data atT= 120 Kbytaking
the population functiong(E‡)= ̄g(Ae−E
‡/(kB× 120 K)
), whereA=5. 0 · 108 s−^1 and
g ̄(x)=
(x/(67 000s−^1 ))^0.^325 e−x/(67 000s− (^1) )
2. 76 kJmole−^1
whenx< 23 kJmole−^1. (9.43)
(The normalization constant has been absorbed intoN 0 .) Above the cutoff energy of 23kJmole−^1 ,
g(x)was taken to be zero. Equation 9.43 gives the curve shown in Figure 9.12b. Substitutingg(E‡)